Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#61095 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#532 ◁ G | |
2790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(4,5)(6,7)(8,9)(10,11)(12,20)(13,16)(14,19)(15,18)(17,23)(21,22);(8,10)(9,14)(11,20)(12,22)(13,18)(15,17)(16,21)(19,23);(5,7)(8,10)(9,15)(11,16)(12,19)(13,18)(14,17)(20,21)(22,23);(8,11,13,15)(9,10,16,18)(12,14,23,21)(17,22,20,19);(9,16)(10,18)(17,20)(19,22)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54083);